The coeeficients c and d are really just one coefficient, because you get a linear relationship between them when you substitute the solution in the equation. To get the other coefficients you need initial conditions.

But what do I put in the "B" bracket? Where does the 'm' come from, and what does it do/why is it there?

You have to substitute

[tex]F_n=\lambda^n[/tex]

in the omogeneous equation, not in the complete one.
You get a polynomial in lambda, that will give various roots, and each of them is a solution. Since the equation is linear, you can multiply each solution by an arbitrary constant and then sum up everything, that's why the "m".
This way you get a solution of the homogeneous equation. To get a complete solution, you have to add another term, that in your particular example must be simply a constant.

But isn't that what I've done by saying that [tex]F_{n} = k^{n}[/tex] for the homogeneous part?

You get a polynomial in lambda, that will give various roots, and each of them is a solution.

Do you mean this? [tex]\sqrt[2p - 5] {2}e^{\frac{2im\pi}{2p - 5}}[/tex] I think it gives the roots for k, but I'm not entirely sure what to do from there except to just assume that whole thing is one root for k.

Since the equation is linear, you can multiply each solution by an arbitrary constant and then sum up everything, that's why the "m".

Okay, that makes sense. I understand this.

This way you get a solution of the homogeneous equation. To get a complete solution, you have to add another term, that in your particular example must be simply a constant.

Is the term that I have to add (17 - 6p)?

Because if we say that the homogeneous part is t, then t = 2t + 6p - 17, so t = 17 - 6p? Or not?